These quests are the same as any other quest, except for a few details. These quests can only be accomplished when you are not in a party. Monster kills done in a party do not count towards the requirements for solo quests.

Solo quests are done sequentially. If you have not completed the Solo quest that comes before the one you want to complete - then the quest will not be available. The solo quests do not generally show up in the quest alarm either.

It is also important to note that all of the solo quest lines merge together at Level 40, with the quest Back to the Original State which is given by Lacriss at Lauke Monastary for which the prize is a Skill Enhancement Stone.